A065775 Array T read by diagonals: T(i,j)=least number of knight's moves on a chessboard (infinite in all directions) needed to move from (0,0) to (i,j).

0, 3, 3, 2, 2, 2, 3, 1, 1, 3, 2, 2, 4, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 2, 2, 2, 4, 4, 5, 3, 3, 3, 3, 3, 3, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 3, 3, 3, 3, 5, 5, 5, 6, 6, 4, 4, 4, 4, 4, 4, 4, 6, 6, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 6, 6, 6, 6, 4, 4, 4, 4, 4, 6, 6, 6, 6, 7, 7, 7, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7

Offset

0,2

Comments

For number of knight's moves to various subsets of the chessboard, see A018837, A183041 - A183053.

Links

Table of n, a(n) for n=0..104.

Formula

T(i,j) is given in cases:

Case 1: row 0

T(0,0)=0, T(1,0)=3, and for m>=1,

T(4m-2,0)=2m, T(4m-1,0)=2m+1, T(4m,0)=2m,

T(4m+1,0)=2m+1.

Case 2: row 1

T(0,1)=3, T(1,1)=2, and for m>=2,

T(4m-2,1)=2m-1, T(4m-1,1)=2m, T(4m,1)=2m+1,

T(4m+1,1)=2m+2.

Case 3: columns 1 and 2

(column 1) = (row 1); (column 2 = row 2).

Case 4: For i>=2 and j>=2,

T(i,j)=1+min{T(i-2,j-1),T(i-1,j-2)}.

Cases 1-4 determine T in the 1st quadrant;

all other T(i,j) are easily obtained by symmetry.

Example

T(i,j) for -2<=i<=2 and -2<=j<=2:
4 1 2 1 4=T(2,2)
1 2 3 2 1=T(2,1)
2 3 0 3 2=T(2,0)
1 2 3 2 1=T(2,-1)
4 1 2 1 4=T(2,-2)
Corner of the array, T(i,j) for i>=0, k>=0:
0 3 2 3 2 3 4...
3 2 1 2 3 4 3...
2 1 4 3 2 3 4...
3 2 3 2 3 4 2...

Crossrefs

Identical to A049604 except for T(1, 1).

Cf. A183041,...,A183042.

Sequence in context: A064983 A124933 A133884 * A096837 A139092 A366424

Adjacent sequences: A065772 A065773 A065774 * A065776 A065777 A065778

Keyword

nonn,tabl

Author

Stewart Gordon, Dec 05 2001

Extensions

Formula, examples, and comments by Clark Kimberling, Dec 20 2010

Example corrected by Clark Kimberling, Oct 14 2016

Status

approved